How many extension dimension of Q field

Question

How many extension dimension of $\Bbb Q(\omega_3,\omega_5)$, $\Bbb Q(2^{1/3},i)$and $\Bbb Q(2^{1/3},\omega_3)$ where $\omega_n=e^{2\pi i/n}$

the first i think 6 extension the second is three third is 4

Answer 1

If $n$ and $m$ are coprime, then $\mathbb{Q}(\omega_n,\omega_m)=\mathbb{Q}(\omega_{nm})$ and the degree is $$ [\mathbb{Q}(\omega_{nm}):\mathbb{Q}]=\phi(nm)=\phi(n)\phi(m). $$ With $n=3, m=5$ we obtain that $[\mathbb{Q}(\omega_3,\omega_5):\mathbb{Q}]=2\cdot 4=8$. For the second one we have $$ [\mathbb{Q}(\sqrt[3]{2},i):\mathbb{Q}]=[\mathbb{Q}(\sqrt[3]{2},i):\mathbb{Q}(\sqrt[3]{2})]\cdot [\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]=2\cdot 3=6. $$ The field is the splitting field of $x^3-2$, as you know - see can we have split field of the real number and how i get the extension dimension?. The last one is for you.